Cross-Multiplication

Date: 08/04/2002 at 01:51:27
From: Cammy
Subject: How do you cross multply?
Please, I know it's simple but I still have trouble.
I can't figure out a way to type it out, so can you just show me the
basics so I can understand?
Thanks, Cammy

Date: 08/04/2002 at 16:14:12
From: Doctor Ian
Subject: Re: How do you cross multply?
Hi Cammy,
When you cross-multiply, you're really multiplying each side by n/n,
where n is the denominator of the other side:
a c
- = -
b d
a d c b
- * - = - * -
b d d b
Note that this doesn't change anything, because multiplying anything
by 1 doesn't change its value. So now we have
ad bc
-- = --
bd bd
Since you're dividing both sides by the same thing, you can ignore
it, to get
ad = bc
Once you get used to the idea, you can skip all the middle steps and
just multiply the things that are crosswise from each other:
a c
. .
. .
- = -
. .
. .
b d
which is where we get the name 'cross-multiplying'.
It's always good to understand something in more than one way, so
here's another way to look at it. When you have any two things that
are equal,
this = that
then if neither of them is zero, it will be true that
this
---- = 1
that
right? Let's look at that for
a c
- = -
b d
a/b
----- = 1
c/d
a d
- * - = 1 Because to divide by a fraction,
b c you invert and multiply...
ad
-- = 1
bc
ad = bc Here, we're just noting that
if this/that = 1, then
this = that...
If you just learn to cross-multiply without understanding _why_ it
works, then it can seem like magic. But after you do a couple of dozen
of these the long way, your brain will automatically start looking
ahead a few steps, and you'll start taking the shortcut naturally,
because you're _ready_ to. And at that point, you'll never really be
able to forget how to cross-multiply, because you'll always be able to
figure it out again if you need to.
In addition, you might want to take a look at this:
Flipping and Switching Fractions
http://mathforum.org/library/drmath/view/58193.html
Does this help?
- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/